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to its adjacent one. To effect this, on the side B'C', equal to BC, construct the regular pentagon B'C'H'I'D′; at the centre of this pentagon, draw a line at right-angles to its plane, and terminating in A', so that B'A'B'C' ; join A'C', A'H', A'I', A'D': the solid angle A' formed by the five planes B'A'C', C'A'H', &c., will be the solid angle required. For the oblique lines A'B', A'C', &c., are equal; one of them A'B'is equal to the side B'C'; hence all the triangles B'A'C', C'A'H', &c., are equal to each other and to the given triangle ABC.

It is farther manifest that the planes B'A'C', C'A'H', &c., are each equally inclined to their adjacent planes; for the solid angles B', C', &c., are all equal, being each formed by two angles of equilateral triangles, and one of a regular pentagon. Let the inclination of the two planes, in which are the equal angles, be named K; the angle K will at the same time be the inclination of each of the planes composing the solid angle A' to their adjacent planes.

This being granted, if at each of the points A, B, C, a solid angle be formed equal to the angle A', we shall have a convex surface DEFG, &c., composed of ten equilateral triangles, every one of which will be inclined to its adjacent triangle by the quantity K; and the angles D, E, F, &c., of

its contour or rim, will alternately combine three angles and two angles of equilateral triangles. Conceive a second surface equal to the surface DEFG, &c.; these two surfaces will adapt themselves to each other, if each triple angle of the one is joined to each double angle of the other; and, since the planes of these angles have already the mutual inclination K, requisite to form a quintuple solid angle equal to the angle A, there will be nothing changed by this junction in the state of either surface, and the two together will form a single continuous surface, composed of twenty equilateral triangles. This surface will be that of the regular icosaedron, since all its solid angles are likewise equal.

Formation of models of the five regular solids.*

The learner will obtain far more definite ideas of the regular solids from models, than from diagrams. These he can easily make for himself from a card or piece of pasteboard.

1. To make the tetraedron. Draw

with care an equilateral triangle on a card, and divide it into four equal equilateral triangles. Trim off the card to the boundary line of the large triangle, and with the point of a knife cut about half through the card on the

line between each of the partial triangles; finally bend the partial triangles back, the part which is not cut serving as a hinge, fold them up so that their edges will touch each other, and you will have a model of the regular solid required. The shaded triangle in the diagram, is designed to represent the base of the solid. The parts may be easily kept in place by touching the edges with a little glue or paste.

* See Appendix to Euclid by John L. Cowley :-also, Hutton's Mathemat ical Recreations, Vol.

2. To make the hexaedron. Draw on a card or pasteboard six equal squares, trim the diagram, cut the lines between them, bend up the parts, &c., as before, and a model of the hexaedron will be completed.

3. To make the octaedron. Draw a diagram containing eight equal and equilateral triangles, cut the boundary lines between them, and proceed as before.

4. To make the

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N. B. In the construction models of the regular solids, it is advısable to make them on a larger scale than the diagrams. An inch or an inch and a half, is a convenient length for the side of the equilateral triangles, the squares and pentagons.

NOTES AND ILLUSTRATIONS.

BOOK I.

DEFINITIONS.

1. GEOMETRY is usually divided into plane and solid geometry. The former investigates the properties of lines, angles and figures which lie in the same plane; the latter treats of solids and lines which lie in different planes. The first five books, with the exception of the third, treat of plane geometry exclusively, while the last three are principally confined to solid geometry.

2.-Geometry is also divided into elementary, and higher geometry. Elementary geometry treats of straight lines, circles, solids bounded by plane surfaces, and the "three round bodies," viz. the cylinder, the cone and the sphere. Higher geometry investigates the properties of spherical triangles, conic sections and other curves.

3. Def. 1.-In the original work of Legendre, that which is said to be the object of geometry to measure, is called “L'étendue.” Dr. Brewster translates the terms by the word space; Professor Farrar, by the word extension. In the text, the word magnitude has been adopted, which, if not more exact, has the advantage of being more readily understood by pupils.

4. Def. 3.-In geometrical figures or diagrams, as well as in drawing and elsewhere, lines are represented by marks or rules, and points by dots, both of which have some breadth, and occupy some space. This is a matter of necessity. For it is obvious that the finest marks that can be drawn upon a blackboard or paper, and the smallest dots that can be made, must unavoidably have some perceptible breadth, and occupy some perceptible space. Were it possible to be otherwise, they would then become invisible and useless. But this imperfection, if imperfection it may be called, does not invali

date the reasoning in the least; and need not lead to any error in calculation or practice.

4. Def. 9.-If two points be taken on the surface of a round body, as a ball, and connected by a straight line, that line will pass through the body, and will not be in its surface.

5. Def. 11.-In the original, an angle is thus defined; "Lorsque deux lignes droites AB, AC, se rencontrent, la quantité plus ou moins grande dont elles sont écartées l'une de l'autre, quant à leur position, s'appelle angle;" &c. Dr. Brewster thus translates it; “When two straight lines AB, AC, meet together, the quantity, greater or less, by which they are separated with regard to their position, is called an angle;" &c.

6. Signs.-The signs used in geometry have been more fully explained than in the original, and the language made to conform to that used in algebra and other branches of mathematics with which the student is already familiar.

8. Axioms.-Legendre has given but five axioms. As there is frequent occasion to refer to others, it is desirable to have them inserted in the work.

9. The words in which a proposition is expressed, are called its enunciation or caption. The former is preferable. Enunciations are particular, or general. Where they refer to some particular diagram they are called particular; in all other cases they are denominated general.

10. The process of reasoning by which propositions are shown to be true, is called demonstration. A demonstration consists in a series of arguments so arranged as to prove the truth or falsity of a theorem, or of the solution of a problem.

A direct demonstration commences with some principles or data which are admitted, or have been proved to be true; and from these, a series of other truths are deduced, each depending on the preceding, till we arrive at the truth which was required to be established. An indirect demonstration is the mode of establishing the truth of a proposition by proving that the supposition of its contrary, involves on an absurdity. This is commonly called reductio ad absurdum. The former is the more common method of conducting a demonstrative argument, and is the most satisfactory to the mind.

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